Problem 2: For an underdamped system (8< wo), the two solutions can be written as x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin w₁t. (a) Show that as →→ wo, x₁(t) approaches the critically damped solution e-ßt. (b) What happens to r₂(t)? Show that the expression x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second critically damped solution te-ßt in the limit 3 → wo.
Problem 2: For an underdamped system (8< wo), the two solutions can be written as x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin w₁t. (a) Show that as →→ wo, x₁(t) approaches the critically damped solution e-ßt. (b) What happens to r₂(t)? Show that the expression x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second critically damped solution te-ßt in the limit 3 → wo.
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Transcribed Image Text:**Problem 2:**
For an underdamped system \((\beta < \omega_0)\), the two solutions can be written as \(x_1(t) = e^{-\beta t} \cos \omega_1 t\) and \(x_2(t) = e^{-\beta t} \sin \omega_1 t\).
(a) Show that as \(\beta \to \omega_0\), \(x_1(t)\) approaches the critically damped solution \(e^{-\beta t}\).
(b) What happens to \(x_2(t)\)? Show that the expression \(x_2(t)/\omega_1\) (which is still a valid solution to the underdamped system) approaches the second critically damped solution \(t e^{-\beta t}\) in the limit \(\beta \to \omega_0\).
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